\(\int (c+d x)^4 \sin ^3(a+b x) \, dx\) [16]

Optimal result

Integrand size = 16, antiderivative size = 225 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=-\frac {488 d^4 \cos (a+b x)}{27 b^5}+\frac {80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}+\frac {8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac {160 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2} \] Output:

-488/27*d^4*cos(b*x+a)/b^5+80/9*d^2*(d*x+c)^2*cos(b*x+a)/b^3-2/3*(d*x+c)^4 
*cos(b*x+a)/b+8/81*d^4*cos(b*x+a)^3/b^5-160/9*d^3*(d*x+c)*sin(b*x+a)/b^4+8 
/3*d*(d*x+c)^3*sin(b*x+a)/b^2+4/9*d^2*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)^2/b^ 
3-1/3*(d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2/b-8/27*d^3*(d*x+c)*sin(b*x+a)^3/b^ 
4+4/9*d*(d*x+c)^3*sin(b*x+a)^3/b^2
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=\frac {-243 \left (24 d^4-12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \cos (a+b x)+\left (8 d^4-36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4\right ) \cos (3 (a+b x))-24 b d (c+d x) \left (242 d^2-39 b^2 (c+d x)^2+\left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{324 b^5} \] Input:

Integrate[(c + d*x)^4*Sin[a + b*x]^3,x]
 

Output:

(-243*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] + ( 
8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b*x)] - 24 
*b*d*(c + d*x)*(242*d^2 - 39*b^2*(c + d*x)^2 + (-2*d^2 + 3*b^2*(c + d*x)^2 
)*Cos[2*(a + b*x)])*Sin[a + b*x])/(324*b^5)
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.34, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 3118}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^4 \sin ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^4 \sin (a+b x)^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int (c+d x)^4 \sin (a+b x)dx+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^4 \sin (a+b x)dx+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \int (c+d x)^3 \cos (a+b x)dx}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \int (c+d x)^3 \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {3 d \int -(c+d x)^2 \sin (a+b x)dx}{b}+\frac {(c+d x)^3 \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \int (c+d x) \cos (a+b x)dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \int -\sin (a+b x)dx}{b}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3118

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin (a+b x)^3dx}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {4 d^2 \left (-\frac {2 d^2 \int \sin ^3(a+b x)dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \left (-\frac {2 d^2 \int \sin (a+b x)^3dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3113

\(\displaystyle -\frac {4 d^2 \left (\frac {2 d^2 \int \left (1-\cos ^2(a+b x)\right )d\cos (a+b x)}{9 b^3}+\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {2 d \int (c+d x) \cos (a+b x)dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {2 d \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {2 d \left (\frac {d \int -\sin (a+b x)dx}{b}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {4 d \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^4 \cos (a+b x)}{b}\right )-\frac {4 d^2 \left (\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

Input:

Int[(c + d*x)^4*Sin[a + b*x]^3,x]
 

Output:

-1/3*((c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]^2)/b + (4*d*(c + d*x)^3*Sin[a 
+ b*x]^3)/(9*b^2) - (4*d^2*((2*d^2*(Cos[a + b*x] - Cos[a + b*x]^3/3))/(9*b 
^3) - ((c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (2*d*(c + d*x)*Sin 
[a + b*x]^3)/(9*b^2) + (2*(-(((c + d*x)^2*Cos[a + b*x])/b) + (2*d*((d*Cos[ 
a + b*x])/b^2 + ((c + d*x)*Sin[a + b*x])/b))/b))/3))/(3*b^2) + (2*(-(((c + 
 d*x)^4*Cos[a + b*x])/b) + (4*d*(((c + d*x)^3*Sin[a + b*x])/b - (3*d*(-((( 
c + d*x)^2*Cos[a + b*x])/b) + (2*d*((d*Cos[a + b*x])/b^2 + ((c + d*x)*Sin[ 
a + b*x])/b))/b))/b))/b))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.80

method	result	size
parallelrisch	\(\frac {\left (27 \left (d x +c \right )^{4} b^{4}-36 d^{2} \left (d x +c \right )^{2} b^{2}+8 d^{4}\right ) \cos \left (3 b x +3 a \right )-36 \left (d x +c \right ) d b \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \sin \left (3 b x +3 a \right )+\left (-243 \left (d x +c \right )^{4} b^{4}+2916 d^{2} \left (d x +c \right )^{2} b^{2}-5832 d^{4}\right ) \cos \left (b x +a \right )+972 \left (d x +c \right ) d b \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \sin \left (b x +a \right )-216 b^{4} c^{4}+2880 b^{2} c^{2} d^{2}-5824 d^{4}}{324 b^{5}}\)	\(181\)
risch	\(-\frac {3 \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \cos \left (b x +a \right )}{4 b^{5}}+\frac {3 d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (b x +a \right )}{b^{4}}+\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-36 b^{2} d^{4} x^{2}-72 b^{2} c \,d^{3} x -36 b^{2} c^{2} d^{2}+8 d^{4}\right ) \cos \left (3 b x +3 a \right )}{324 b^{5}}-\frac {d \left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \sin \left (3 b x +3 a \right )}{27 b^{4}}\)	\(344\)
orering	\(\frac {16 d \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}-891 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x -3564 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}-5346 b^{4} c^{2} d^{4} x^{2}-3564 b^{4} c^{3} d^{3} x -891 b^{4} c^{4} d^{2}-1960 b^{2} d^{6} x^{2}-3920 b^{2} c \,d^{5} x -1960 b^{2} c^{2} d^{4}+5460 d^{6}\right ) \sin \left (b x +a \right )^{3}}{243 b^{8} \left (d x +c \right )^{3}}-\frac {2 \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}-396 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x -1584 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}-2376 b^{4} c^{2} d^{4} x^{2}-1584 b^{4} c^{3} d^{3} x -396 b^{4} c^{4} d^{2}-10400 b^{2} d^{6} x^{2}-20800 b^{2} c \,d^{5} x -10400 b^{2} c^{2} d^{4}+21840 d^{6}\right ) \left (4 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{4} \sin \left (b x +a \right )^{2} b \cos \left (b x +a \right )\right )}{243 b^{8} \left (d x +c \right )^{6}}+\frac {16 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}-105 b^{2} d^{4} x^{2}-210 b^{2} c \,d^{3} x -105 b^{2} c^{2} d^{2}+182 d^{4}\right ) \left (12 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{3} d^{2}+24 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{2} d b \cos \left (b x +a \right )+6 \left (d x +c \right )^{4} \sin \left (b x +a \right ) b^{2} \cos \left (b x +a \right )^{2}-3 \left (d x +c \right )^{4} \sin \left (b x +a \right )^{3} b^{2}\right )}{81 b^{8} \left (d x +c \right )^{5}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-360 b^{2} d^{4} x^{2}-720 b^{2} c \,d^{3} x -360 b^{2} c^{2} d^{2}+728 d^{4}\right ) \left (24 \left (d x +c \right ) \sin \left (b x +a \right )^{3} d^{3}+108 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{2} d^{2} b \cos \left (b x +a \right )+72 \left (d x +c \right )^{3} \sin \left (b x +a \right ) d \,b^{2} \cos \left (b x +a \right )^{2}-36 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{4} b^{3} \cos \left (b x +a \right )^{3}-21 \left (d x +c \right )^{4} \sin \left (b x +a \right )^{2} b^{3} \cos \left (b x +a \right )\right )}{243 b^{8} \left (d x +c \right )^{4}}\)	\(883\)
derivativedivides	\(\text {Expression too large to display}\)	\(1023\)
default	\(\text {Expression too large to display}\)	\(1023\)

Input:

int((d*x+c)^4*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/324*((27*(d*x+c)^4*b^4-36*d^2*(d*x+c)^2*b^2+8*d^4)*cos(3*b*x+3*a)-36*(d* 
x+c)*d*b*((d*x+c)^2*b^2-2/3*d^2)*sin(3*b*x+3*a)+(-243*(d*x+c)^4*b^4+2916*d 
^2*(d*x+c)^2*b^2-5832*d^4)*cos(b*x+a)+972*(d*x+c)*d*b*((d*x+c)^2*b^2-6*d^2 
)*sin(b*x+a)-216*b^4*c^4+2880*b^2*c^2*d^2-5824*d^4)/b^5
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.56 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 252 \, b^{2} c^{2} d^{2} + 488 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 14 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 14 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right ) + 12 \, {\left (21 \, b^{3} d^{4} x^{3} + 63 \, b^{3} c d^{3} x^{2} + 21 \, b^{3} c^{3} d - 122 \, b c d^{3} - {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + {\left (63 \, b^{3} c^{2} d^{2} - 122 \, b d^{4}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \] Input:

integrate((d*x+c)^4*sin(b*x+a)^3,x, algorithm="fricas")
 

Output:

1/81*((27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 36*b^2*c^2*d^2 + 
8*d^4 + 18*(9*b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d - 2*b^2*c*d^3 
)*x)*cos(b*x + a)^3 - 3*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 
 252*b^2*c^2*d^2 + 488*d^4 + 18*(9*b^4*c^2*d^2 - 14*b^2*d^4)*x^2 + 36*(3*b 
^4*c^3*d - 14*b^2*c*d^3)*x)*cos(b*x + a) + 12*(21*b^3*d^4*x^3 + 63*b^3*c*d 
^3*x^2 + 21*b^3*c^3*d - 122*b*c*d^3 - (3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3 
*b^3*c^3*d - 2*b*c*d^3 + (9*b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^2 + (63 
*b^3*c^2*d^2 - 122*b*d^4)*x)*sin(b*x + a))/b^5
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (226) = 452\).

Time = 0.67 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.43 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**4*sin(b*x+a)**3,x)
 

Output:

Piecewise((-c**4*sin(a + b*x)**2*cos(a + b*x)/b - 2*c**4*cos(a + b*x)**3/( 
3*b) - 4*c**3*d*x*sin(a + b*x)**2*cos(a + b*x)/b - 8*c**3*d*x*cos(a + b*x) 
**3/(3*b) - 6*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)/b - 4*c**2*d**2* 
x**2*cos(a + b*x)**3/b - 4*c*d**3*x**3*sin(a + b*x)**2*cos(a + b*x)/b - 8* 
c*d**3*x**3*cos(a + b*x)**3/(3*b) - d**4*x**4*sin(a + b*x)**2*cos(a + b*x) 
/b - 2*d**4*x**4*cos(a + b*x)**3/(3*b) + 28*c**3*d*sin(a + b*x)**3/(9*b**2 
) + 8*c**3*d*sin(a + b*x)*cos(a + b*x)**2/(3*b**2) + 28*c**2*d**2*x*sin(a 
+ b*x)**3/(3*b**2) + 8*c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**2/b**2 + 28* 
c*d**3*x**2*sin(a + b*x)**3/(3*b**2) + 8*c*d**3*x**2*sin(a + b*x)*cos(a + 
b*x)**2/b**2 + 28*d**4*x**3*sin(a + b*x)**3/(9*b**2) + 8*d**4*x**3*sin(a + 
 b*x)*cos(a + b*x)**2/(3*b**2) + 28*c**2*d**2*sin(a + b*x)**2*cos(a + b*x) 
/(3*b**3) + 80*c**2*d**2*cos(a + b*x)**3/(9*b**3) + 56*c*d**3*x*sin(a + b* 
x)**2*cos(a + b*x)/(3*b**3) + 160*c*d**3*x*cos(a + b*x)**3/(9*b**3) + 28*d 
**4*x**2*sin(a + b*x)**2*cos(a + b*x)/(3*b**3) + 80*d**4*x**2*cos(a + b*x) 
**3/(9*b**3) - 488*c*d**3*sin(a + b*x)**3/(27*b**4) - 160*c*d**3*sin(a + b 
*x)*cos(a + b*x)**2/(9*b**4) - 488*d**4*x*sin(a + b*x)**3/(27*b**4) - 160* 
d**4*x*sin(a + b*x)*cos(a + b*x)**2/(9*b**4) - 488*d**4*sin(a + b*x)**2*co 
s(a + b*x)/(27*b**5) - 1456*d**4*cos(a + b*x)**3/(81*b**5), Ne(b, 0)), ((c 
**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x**3 + c*d**3*x**4 + d**4*x**5/5)*sin( 
a)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (205) = 410\).

Time = 0.09 (sec) , antiderivative size = 934, normalized size of antiderivative = 4.15 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*sin(b*x+a)^3,x, algorithm="maxima")
 

Output:

1/324*(108*(cos(b*x + a)^3 - 3*cos(b*x + a))*c^4 - 432*(cos(b*x + a)^3 - 3 
*cos(b*x + a))*a*c^3*d/b + 648*(cos(b*x + a)^3 - 3*cos(b*x + a))*a^2*c^2*d 
^2/b^2 - 432*(cos(b*x + a)^3 - 3*cos(b*x + a))*a^3*c*d^3/b^3 + 108*(cos(b* 
x + a)^3 - 3*cos(b*x + a))*a^4*d^4/b^4 + 36*(3*(b*x + a)*cos(3*b*x + 3*a) 
- 27*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) + 27*sin(b*x + a))*c^3*d/b 
- 108*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a) - sin(3*b* 
x + 3*a) + 27*sin(b*x + a))*a*c^2*d^2/b^2 + 108*(3*(b*x + a)*cos(3*b*x + 3 
*a) - 27*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) + 27*sin(b*x + a))*a^2* 
c*d^3/b^3 - 36*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a) - 
 sin(3*b*x + 3*a) + 27*sin(b*x + a))*a^3*d^4/b^4 + 18*((9*(b*x + a)^2 - 2) 
*cos(3*b*x + 3*a) - 81*((b*x + a)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin(3* 
b*x + 3*a) + 162*(b*x + a)*sin(b*x + a))*c^2*d^2/b^2 - 36*((9*(b*x + a)^2 
- 2)*cos(3*b*x + 3*a) - 81*((b*x + a)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*si 
n(3*b*x + 3*a) + 162*(b*x + a)*sin(b*x + a))*a*c*d^3/b^3 + 18*((9*(b*x + a 
)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x + a)^2 - 2)*cos(b*x + a) - 6*(b*x + a 
)*sin(3*b*x + 3*a) + 162*(b*x + a)*sin(b*x + a))*a^2*d^4/b^4 + 12*(3*(3*(b 
*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x + 3*a) - 81*((b*x + a)^3 - 6*b*x - 6*a) 
*cos(b*x + a) - (9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) + 243*((b*x + a)^2 - 
2)*sin(b*x + a))*c*d^3/b^3 - 12*(3*(3*(b*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x 
 + 3*a) - 81*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) - (9*(b*x + a)^2 ...
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.56 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \cos \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} - \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (b x + a\right )}{4 \, b^{5}} - \frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac {3 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (b x + a\right )}{b^{5}} \] Input:

integrate((d*x+c)^4*sin(b*x+a)^3,x, algorithm="giac")
 

Output:

1/324*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 108*b^4* 
c^3*d*x + 27*b^4*c^4 - 36*b^2*d^4*x^2 - 72*b^2*c*d^3*x - 36*b^2*c^2*d^2 + 
8*d^4)*cos(3*b*x + 3*a)/b^5 - 3/4*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c 
^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 12*b^2*d^4*x^2 - 24*b^2*c*d^3*x - 1 
2*b^2*c^2*d^2 + 24*d^4)*cos(b*x + a)/b^5 - 1/27*(3*b^3*d^4*x^3 + 9*b^3*c*d 
^3*x^2 + 9*b^3*c^2*d^2*x + 3*b^3*c^3*d - 2*b*d^4*x - 2*b*c*d^3)*sin(3*b*x 
+ 3*a)/b^5 + 3*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3* 
d - 6*b*d^4*x - 6*b*c*d^3)*sin(b*x + a)/b^5
 

Mupad [B] (verification not implemented)

Time = 36.90 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.37 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=\frac {8\,x\,{\cos \left (a+b\,x\right )}^3\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^3}-\frac {2\,{\cos \left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4-360\,b^2\,c^2\,d^2+728\,d^4\right )}{81\,b^5}-\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (27\,b^4\,c^4-252\,b^2\,c^2\,d^2+488\,d^4\right )}{27\,b^5}-\frac {8\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^4}-\frac {2\,d^4\,x^4\,{\cos \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,{\sin \left (a+b\,x\right )}^3\,\left (122\,c\,d^3-21\,b^2\,c^3\,d\right )}{27\,b^4}+\frac {28\,d^4\,x^3\,{\sin \left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,x\,{\sin \left (a+b\,x\right )}^3\,\left (122\,d^4-63\,b^2\,c^2\,d^2\right )}{27\,b^4}+\frac {4\,x^2\,{\cos \left (a+b\,x\right )}^3\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^3}+\frac {2\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{3\,b^3}-\frac {8\,c\,d^3\,x^3\,{\cos \left (a+b\,x\right )}^3}{3\,b}-\frac {d^4\,x^4\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b}+\frac {8\,d^4\,x^3\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^2}+\frac {28\,c\,d^3\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}-\frac {8\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^4}+\frac {4\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{3\,b^3}-\frac {4\,c\,d^3\,x^3\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b}+\frac {8\,c\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2} \] Input:

int(sin(a + b*x)^3*(c + d*x)^4,x)
 

Output:

(8*x*cos(a + b*x)^3*(20*c*d^3 - 3*b^2*c^3*d))/(9*b^3) - (2*cos(a + b*x)^3* 
(728*d^4 + 27*b^4*c^4 - 360*b^2*c^2*d^2))/(81*b^5) - (cos(a + b*x)*sin(a + 
 b*x)^2*(488*d^4 + 27*b^4*c^4 - 252*b^2*c^2*d^2))/(27*b^5) - (8*cos(a + b* 
x)^2*sin(a + b*x)*(20*c*d^3 - 3*b^2*c^3*d))/(9*b^4) - (2*d^4*x^4*cos(a + b 
*x)^3)/(3*b) - (4*sin(a + b*x)^3*(122*c*d^3 - 21*b^2*c^3*d))/(27*b^4) + (2 
8*d^4*x^3*sin(a + b*x)^3)/(9*b^2) - (4*x*sin(a + b*x)^3*(122*d^4 - 63*b^2* 
c^2*d^2))/(27*b^4) + (4*x^2*cos(a + b*x)^3*(20*d^4 - 9*b^2*c^2*d^2))/(9*b^ 
3) + (2*x^2*cos(a + b*x)*sin(a + b*x)^2*(14*d^4 - 9*b^2*c^2*d^2))/(3*b^3) 
- (8*c*d^3*x^3*cos(a + b*x)^3)/(3*b) - (d^4*x^4*cos(a + b*x)*sin(a + b*x)^ 
2)/b + (8*d^4*x^3*cos(a + b*x)^2*sin(a + b*x))/(3*b^2) + (28*c*d^3*x^2*sin 
(a + b*x)^3)/(3*b^2) - (8*x*cos(a + b*x)^2*sin(a + b*x)*(20*d^4 - 9*b^2*c^ 
2*d^2))/(9*b^4) + (4*x*cos(a + b*x)*sin(a + b*x)^2*(14*c*d^3 - 3*b^2*c^3*d 
))/(3*b^3) - (4*c*d^3*x^3*cos(a + b*x)*sin(a + b*x)^2)/b + (8*c*d^3*x^2*co 
s(a + b*x)^2*sin(a + b*x))/b^2
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.72 \[ \int (c+d x)^4 \sin ^3(a+b x) \, dx=\frac {36 \sin \left (b x +a \right )^{3} b^{3} c^{3} d +36 \sin \left (b x +a \right )^{3} b^{3} d^{4} x^{3}-24 \sin \left (b x +a \right )^{3} b c \,d^{3}-24 \sin \left (b x +a \right )^{3} b \,d^{4} x +216 a \,b^{3} c^{3} d -576 a b c \,d^{3}-1456 \cos \left (b x +a \right ) d^{4}-496 d^{4}-54 \cos \left (b x +a \right ) b^{4} c^{4}-54 \cos \left (b x +a \right ) b^{4} d^{4} x^{4}+720 \cos \left (b x +a \right ) b^{2} c^{2} d^{2}+720 \cos \left (b x +a \right ) b^{2} d^{4} x^{2}+216 \sin \left (b x +a \right ) b^{3} c^{3} d +216 \sin \left (b x +a \right ) b^{3} d^{4} x^{3}-1440 \sin \left (b x +a \right ) b c \,d^{3}-1440 \sin \left (b x +a \right ) b \,d^{4} x -27 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{4} c^{4}+72 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{2} c \,d^{3} x -8 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} d^{4}+288 b^{2} c^{2} d^{2}-27 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{4} d^{4} x^{4}+36 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{2} c^{2} d^{2}+36 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{2} d^{4} x^{2}+108 \sin \left (b x +a \right )^{3} b^{3} c^{2} d^{2} x +108 \sin \left (b x +a \right )^{3} b^{3} c \,d^{3} x^{2}-216 \cos \left (b x +a \right ) b^{4} c^{3} d x -324 \cos \left (b x +a \right ) b^{4} c^{2} d^{2} x^{2}-216 \cos \left (b x +a \right ) b^{4} c \,d^{3} x^{3}+1440 \cos \left (b x +a \right ) b^{2} c \,d^{3} x +648 \sin \left (b x +a \right ) b^{3} c^{2} d^{2} x +648 \sin \left (b x +a \right ) b^{3} c \,d^{3} x^{2}-54 b^{4} c^{4}-108 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{4} c^{3} d x -162 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{4} c^{2} d^{2} x^{2}-108 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{4} c \,d^{3} x^{3}}{81 b^{5}} \] Input:

int((d*x+c)^4*sin(b*x+a)^3,x)
 

Output:

( - 27*cos(a + b*x)*sin(a + b*x)**2*b**4*c**4 - 108*cos(a + b*x)*sin(a + b 
*x)**2*b**4*c**3*d*x - 162*cos(a + b*x)*sin(a + b*x)**2*b**4*c**2*d**2*x** 
2 - 108*cos(a + b*x)*sin(a + b*x)**2*b**4*c*d**3*x**3 - 27*cos(a + b*x)*si 
n(a + b*x)**2*b**4*d**4*x**4 + 36*cos(a + b*x)*sin(a + b*x)**2*b**2*c**2*d 
**2 + 72*cos(a + b*x)*sin(a + b*x)**2*b**2*c*d**3*x + 36*cos(a + b*x)*sin( 
a + b*x)**2*b**2*d**4*x**2 - 8*cos(a + b*x)*sin(a + b*x)**2*d**4 - 54*cos( 
a + b*x)*b**4*c**4 - 216*cos(a + b*x)*b**4*c**3*d*x - 324*cos(a + b*x)*b** 
4*c**2*d**2*x**2 - 216*cos(a + b*x)*b**4*c*d**3*x**3 - 54*cos(a + b*x)*b** 
4*d**4*x**4 + 720*cos(a + b*x)*b**2*c**2*d**2 + 1440*cos(a + b*x)*b**2*c*d 
**3*x + 720*cos(a + b*x)*b**2*d**4*x**2 - 1456*cos(a + b*x)*d**4 + 36*sin( 
a + b*x)**3*b**3*c**3*d + 108*sin(a + b*x)**3*b**3*c**2*d**2*x + 108*sin(a 
 + b*x)**3*b**3*c*d**3*x**2 + 36*sin(a + b*x)**3*b**3*d**4*x**3 - 24*sin(a 
 + b*x)**3*b*c*d**3 - 24*sin(a + b*x)**3*b*d**4*x + 216*sin(a + b*x)*b**3* 
c**3*d + 648*sin(a + b*x)*b**3*c**2*d**2*x + 648*sin(a + b*x)*b**3*c*d**3* 
x**2 + 216*sin(a + b*x)*b**3*d**4*x**3 - 1440*sin(a + b*x)*b*c*d**3 - 1440 
*sin(a + b*x)*b*d**4*x + 216*a*b**3*c**3*d - 576*a*b*c*d**3 - 54*b**4*c**4 
 + 288*b**2*c**2*d**2 - 496*d**4)/(81*b**5)